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Well, actually it is "either, or." I don't like either chocolate or ice cream. If you leave out the don't it would be I like neither chocolate nor ice cream. (A double negative always makes a positive, just like in math.) The and gets tricky because it becomes ambiguous...I don't like chocolate and ice cream can very well mean the two together.

So. If you were to walk up to a deli counter and ask for a ham and swiss sandwich, and they responded, "Sorry, I'm don't have any ham and swiss," it would still be valid to ask for a ham and cheddar. However, if they said "Sorry, I don't have any ham OR swiss," then ham and cheddar would be right out.

By the same token, "I didn't sleep last night and the night before" means one could have slept on but one of either night, just as the lack of "ham and swiss" does not necessarily preculde both ingredients individually.

Jon is correct. Andrew is making an error (or, at least, an implausible assumption) about the scope of the operators "and" and "or" (i.e., about the bits of the sentence they are supposed to be conjoining or disjoining).*

Andrew seems to be reading the sentences as if they were "I didn't sleep last night and/or I didn't sleep the night before," or, using brackets to make the scope more obvious, "[I didn't sleep last night] AND/OR [I didn't sleep the night before]." But the sentences as given are much more naturally analyzed as follows: "I didn't sleep [last night AND/OR the night before]." If the intended meaning is that the speaker did not sleep on either night, she should say either "I didn't sleep last night or the night before," or "I didn't sleep last night and I didn't sleep the night before."

To avoid all chance of scope confusion, but at the cost of sounding dreadfully stilted, she could say, "I slept neither last night not the night before."

*In logical parlance, "and" is the conjunction operator; "or" is the disjunction operator.

In logic, "not (A or B)" is the same as "(not A) and (not B)"Similarly, "not (A and B)" is the same as "(not A) or (not B)"

You can apply these De Morgan laws to English sentences.

But first, the meaning of the sentence has to be clear.

"I don't like A and B" can have several different meanings: "(I don't like A) and (I don't like B)" "I don't like C" (where C is the combination of A and B)

If the first meaning is the intended meaning, then you might argue that these statements are the same:"I don't like A and B""(I don't like A) and (I don't like B)""not (I like A) and not (I like B)""not (I like A or I like B)""not (I like A or B)""I don't like A or B"

But notice how the first and the last statement are different? Where did exactly did this transformation take place?

The answer is that the first and the last sentences have multiple meanings.

However, these more precise statements are the same:"(I don't like A) and (I don't like B)""I don't like (A or B)"

Whereas these two are different:"(I don't like A) and (I don't like B)""(I don't like A) or (I don't like B)"

This is the heart of the problem. English is ambiguous. The intended meaning can usually be inferred by a native speaker. But a non-native speaker might associate a different logical meaning with the same sentence. That's where the confusion lies.

But there are other, smaller complications as well:

It's not necessarily true that "I don't like A" is the same as "not (I like A)." You may be indifferent to A. In that case, "I don't like A" is false, but "not (I like A)" is true. So in reality, the two statements are not necessarily the same.

The reason for this is the third value--indifference--in addition to like and dislike: This makes it a trinary logic system, rather than the usual true or false binary logic system. In real life, there may be logic systems like this.

Another complication is that the English phrase "A or B" can mean "A or B or both." But the same English phrase can also mean "either A, or B, but not both."

For example:"Clean your room or go to bed now!"

The "or" in that sentence has a different meaning than the "or" in this sentence:"I would like some ice cream or chocolate."

Another consideration is that sometimes the same English words can be used to describe things that are different. For example, someone may like chocolate and ice cream (A and B) when it's ice cream with a little chocolate swirled on top, but would NOT like chocolate and ice cream (A and B) when it's essentially a bowl of chocolate with just a spoonful of ice cream on top.

If there is a distinction between these two cases, then the statement "A and B" is not precise enough. It needs to be clarified further in order to understand what was intended.

English speakers usually able to infer the intended meaning. But it can happen that the same words can be interpreted differently, and when this happens, it is a major source of confusion. Many "paradoxes" are based on this sort of thing.

Your OR definitely sounds better. I'm no expert either, but I think in these instances it would be NOR.

It's a negative context (didn't and don't). To use one of your examples, if I were to draw out the sentence, I would say,

"I don't like chocolate, nor do I like ice cream."

So to shorten it, would be, "I don't like chocolate, nor ice cream."

To use AND doesn't work for me because it seems like I'd be saying I don't like chocolate and ice cream TOGETHER. In other words, I like ice cream and I like chocolate, but I don't like chocolate ice cream.

But your wife saying, "I didn't sleep last night AND the night before" is different. You could say it this way and it would make sense:

"I didn't sleep last night AND I didn't sleep the night before EITHER."

This is an interesting question. As a native speaker, my first thought on hearing "I didn't sleep last night and the night before" would be that the speaker meant to continue the sentence and say something like "I didn't sleep last night and the night before somebody kept waking me." The reason to use "or" here is that it makes a stronger connection.

But my first thought on hearing "I don't like chocolate and ice cream" would be that the speaker disliked having chocolate on her ice cream. Maybe she likes them separately, but she certainly doesn't like them together. The reason to use "or" here is that it makes a weaker connection.

Regarding the previous comment: "I don't like chocolate, nor ice cream" is jarring to me. As a native speaker I would either say: I don't like chocolate or ice cream" or "I don't like chocolate nor do I like ice cream," but I would never mix them.

But I think the commenter is on the right track. My guess is that these odd structures are survivals of the old double negative. Double negatives were very common in English before the seventeenth century. Probably back then people would say something like:

"I didn't neither sleep last night nor the night before."

"I don't like neither chocolate nor ice cream."

And when the double negative went out of favor, the sentence kept its shape anyway.

I tried to find a good example in Shakespeare. The best I could come up with was from Twelfth Night:

"She'll not match above her degree, neither in estate, years, nor wit."

Modernized:

"She won't marry above her station in estate, years, or wit."

Probably there's a better explanation of this out there on the Internet somewhere, but I can't find it.

Sidebar on the 12th night quote. I don't see a double negative there. It's practically two statements, the first broad, the second more a refined, and from the speaker's point of view, a more authoritative and witty qualification. Without the second negative, which Paul drops in the modernization, we lose the sense that the speaker is pausing and making a finer point. By the way, I get why she'd want to marry above her degree in estate and wit, but why years? Must be a patriarchal thing.

When I read the statement, "I didn't sleep last night AND the night before," I think to myself, "and... and the night before WHAT? What about the night before?"

The "and" separates the statement into two parts. It would be correct to make a statement like, "I didn't take out the trash and the recycling is piling up." That's because "and" is additive, and a negative statement is subtractive. "I like this and this" states that there is a group of things you like. "I dislike this and this" also works, even though it's the opposite, because you're making an inclusive statement of the things described as "what I dislike." "I do not like this or this" is correct because it is a negative statement pointing out what you don't do.

"I was sleepless last night AND the night before" is correct for this reason - "and" includes the two nights in the group of things described as "sleepless."

"I did not sleep last night OR the night before" is correct."I did not sleep last night AND the night before" is incorrect.

Technically speaking, in formal logic, the statement "I didn't sleep last night and the night before" cold be true if you had slept on either one but not both of those nights, as you did not sleep on both last night and the night before.

Last night OR the night before does in fact state that one slept neither last night nor the night before.

I stumbled this page. This site is fantastic! I long for debates like this in my everyday life. "This sentence doesn't make sense but the reason it doesn't make sense is mysterious and intriguing; let's talk about it!"

I've found this direction of conversation has always been met with an expression that can only be described as a mixture of terror and despondence. I'm so desperate, however, that I choose to ignore my friends' unmistakable reactions and continue, attempting to spike the conversation with offhand references to monster trucks and explosions.

I came to learn that as languages evolve, and especially the English language, they tend to get easier to speak. For instance: UK English still speaks in the present perfect, whereas, American English speak in simple. My point is two-fold. When I went to school to learn how to teach English as a second language I learned that as long as one is communicating in a rather clear and concise manner, one really shouldn't get too hung up on the grammar rules. I for one really struggled with that because I am a heavy punctuator. Getting back to how languages evolve, they tend to evolve merely in what simply just sounds better, and if we really want to answer the question that was presented the answer is really simple:

Not only is it correct to say "or" but it sounds better, thus: I DIDN'T SLEEP LAST NIGHT OR THE NIGHT BEFORE, is correct. However, if she clearly communicated to you that neither night was one filled with sleep, and especially as a foreigner to our English tongue, then either way is correct, because when it comes down to it, communication is the intent. :)

Lada mentioned using "or" in a negative sentence, and "and" in a positive sentence.

I like chocolate and vanilla ice cream.

I don't like chocolate or vanilla ice cream.

However, you could very well say "I like chocolate or vanilla ice cream" in response to someone asking you which you like - whether as a general question, or asking for the person's order/decision on which ice cream flavor (s)he prefers.

In this case, the person is saying "I could go for either (the flavor) chocolate or vanilla for my ice cream. Either is fine with me, you can choose"... so to say.

In the second sentence Lada wrote, using "or", "I don't like chocolate or vanilla ice cream".

As others have mentioned before, you very well could say "I don't like chocolate and vanilla ice cream", which could be taken that the speaker doesn't like chocolate and vanilla ice cream TOGETHER.